# Factors with only two variables in factor analysis

I am running a factor analysis and have a couple of questions.

I have 10 variables, all of them come from a survey, with each answer is in the scale of 1 to 7. I have calculated a correlation matrix between the 10 variables, and looked for correlations of 0.5 or higher. These made me notice of 4 groups of correlated variables, two groups of 3 variables and two groups of 2 variables.

Then I ran the factor analysis. According to the scree plot, I should take 4 factors. According to the eigenvalues, the 4th eigenvalue is 1.006, so it's slightly on the line, it is larger than 1, but not by much...In addition, after 4 factor the cumulative variance is 82%. After 3 it is only 72%.

My question is, should I choose 4 factors or 3 ? Is it OK to have a factor with only two variables that construct it ? I know that a single variable factor is something we don't want, just like we don't want a sample-specific factors. But what is the rule regarding two variables in a factor ? I looked on the net, and found one document saying the minimum is 3. I didn't see any mathematical explanation and I am not convinced. What can you recommend me ?

• Scree and the K1 method seem suggest that you could retain the 4th factor. Maybe clarifying the ultimate goal of your analysis and setting a performance metric for your modeling could help deciding if you ultimately should include the 4th factor. Apr 23, 2014 at 13:31
• A rule of thumb motivated by some properties of FA is "3+ variableds on a factor". So I would recommed you 3 factors. But that is not a law. If you see 4 factors reproduce correlations much better and the factors are better interpretable, choose 4. Then I'd recommend not to interpret the factor with 2 items. Try to get more items to support it (if that is possible) and redo FA. Jan 11, 2017 at 17:16
• Given observations (1 observation = 1 vector of variable values) and you want to find the relationships between the 10 variables, I suggest Greg Ver Steeg CorEx method. You can start with the linear sieve which is fast. isi.edu/projects/gregv/correlation_explanation The python package(s) also offer nice visualization of dependencies. Jan 5, 2018 at 4:03
• Also I caution you against relying on the "eigenvalue >1" rule. See any source using the phrase 'tom swift's electric factor analysis machine'. Apr 11, 2018 at 14:05